The indicator function of a convex set $C\subset N_{\mathbb{R}}$ is the concave function $\iota_{C}$ defined as $\iota_{C}(u)=0$ for $u\in C$ and $\iota_{C}(u)=-\infty$ for $u\not\in C$. Observe that $\iota_{C}$ is the logarithm of the characteristic function of $C$. This function is closed if and only if $C$ is a closed set.

The support function of a convex set $C$ is the function

It is a closed concave function. A function $f\colon M_{\mathbb{R}}\to\mathbb{R}$ is called conical if $f(\lambda x)=\lambda f(x)$ for all $\lambda\geq 0$. The support function $\Psi_{C}$ is conical. The converse is also true: all conical closed concave functions are of the form $\Psi_{C}$ for a closed convex set $C$.

We have $\iota_{C}^{\vee}=\Psi_{C}$ and $\Psi_{C}^{\vee}={\operatorname{cl}}(\iota_{C})=\iota_{{\overline{C}}}$. Thus, the Legendre-Fenchel duality defines a bijective correspondence between indicator functions of closed convex subsets of $N_{\mathbb{R}}$ and closed concave conical functions on $M_{\mathbb{R}}$.